Best Known (40, 40+39, s)-Nets in Base 64
(40, 40+39, 513)-Net over F64 — Constructive and digital
Digital (40, 79, 513)-net over F64, using
- t-expansion [i] based on digital (28, 79, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(40, 40+39, 514)-Net in Base 64 — Constructive
(40, 79, 514)-net in base 64, using
- 1 times m-reduction [i] based on (40, 80, 514)-net in base 64, using
- base change [i] based on digital (20, 60, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 20, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- digital (0, 40, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 20, 257)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (20, 60, 514)-net over F256, using
(40, 40+39, 1637)-Net over F64 — Digital
Digital (40, 79, 1637)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6479, 1637, F64, 2, 39) (dual of [(1637, 2), 3195, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6479, 2052, F64, 2, 39) (dual of [(2052, 2), 4025, 40]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6479, 4104, F64, 39) (dual of [4104, 4025, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(35) [i] based on
- linear OA(6477, 4096, F64, 39) (dual of [4096, 4019, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(6471, 4096, F64, 36) (dual of [4096, 4025, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(38) ⊂ Ce(35) [i] based on
- OOA 2-folding [i] based on linear OA(6479, 4104, F64, 39) (dual of [4104, 4025, 40]-code), using
- discarding factors / shortening the dual code based on linear OOA(6479, 2052, F64, 2, 39) (dual of [(2052, 2), 4025, 40]-NRT-code), using
(40, 40+39, 3271280)-Net in Base 64 — Upper bound on s
There is no (40, 79, 3271281)-net in base 64, because
- 1 times m-reduction [i] would yield (40, 78, 3271281)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 762 146490 572564 939077 256531 711063 201317 030224 977449 731079 854621 463590 041836 721894 489327 682020 354489 368258 264933 904822 451259 799280 969293 830048 > 6478 [i]